The drawdown is the measure of the decline from a historical peak in some variable. Somewhat more formally, if is a stochastic process with, the drawdown at time, denoted, is defined as:The average drawdown up to time is the time average of drawdowns that have occurred up to time :The maximum drawdown up to time is the maximum of the drawdown over the history of the variable. More formally, the MDD is defined as:
Pseudocode
The following pseudocode computes the Drawdown and Max Drawdown of the variable "NAV", the Net Asset Value of an investment. Drawdown and Max Drawdown are calculated as percentages: MDD = 0 peak = -99999 for i = 1 to N step 1do # peak will be the maximum value seen so far, only get updated when higher NAV is seen ifthen peak = NAV end if DD = 100.0 × / peak # Same idea as peak variable, MDD keeps track of the maximum drawdown so far. Only get updated when higher DD is seen. ifthen MDD = DD end if end for
In finance, the use of the maximum drawdown as an indicator of risk is particularly popular in the world of commodity trading advisors through the widespread use of three performance measures: the Calmar ratio, the Sterling ratio and the Burke ratio. These measures can be considered as a modification of the Sharpe ratio in the sense that the numerator is always the excess of mean returns over the risk-free rate while the standard deviation of returns in the denominator is replaced by some function of the drawdown.
2. How long it lasts (the duration)
Many assume Max DD Duration is the length of time between new highs during which the Max DD occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss, but it doesn’t have to be. When is Brownian motion with drift, the expected behavior of the MDD as a function of time is known. If is represented as:Where is a standard Wiener process, then there are three possible outcomes based on the behavior of the drift :
implies that the MDD grows logarithmically with time
Where an amount of credit is offered, a drawdown against the line of credit results in a debt
Funds offered
Where funds are made available, such as for a specific purpose, drawdowns occur if the funds – or a portion of the funds – are released when conditions are met.
Optimization of drawdown
A passing glance at the mathematical definition of drawdown suggests significant difficulty in using an optimization framework to minimize the quantity, subject to other constraints; this is due to the non-convex nature of the problem. However, there is a way to turn the drawdown minimization problem into a linear program. The authors start by proposing an auxiliary function, where is a vector of portfolio returns, that is defined by:They call this the conditional drawdown-at-risk ; this is a nod to conditional value-at-risk, which may also be optimized using linear programming. There are two limiting cases to be aware of:
